Stalks

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 16 November 2009

Stalks

Let $X$ be a topological space, let $x\in X$ and let $ℱ:𝒯\to 𝒜$ be a functor.

Let ${𝒩}_X$ be the neighourhood filter at $x$.

The stalk of $𝒰$ at $x$ is $${ℱ}_x=\underset{U\in {𝒩}_x}{\lim }ℱ\left(U\right)\text{(direct limit).}$$

Let $$E=\prod _{x\in X}{ℱ}_x$$ with topology given by setting $W$ open if it satisfies the following constraint. $$\text{If}U\in 𝒯\text{and}s\in ℱ\left(U\right)\text{then}{\tilde{s}}^{-1}\left(W\right)\text{is open,}$$ where $$\begin{array}{rcl}\tilde{s}:U& ⟶& E\\ x& ⟼& s_x\end{array}\text{is given by}\begin{array}{rcl}ℱ\left(U\right)& ⟶& {ℱ}_x\\ s& ⟼& s_x\end{array}$$ is the natural homomorphism.

The sheafification of $ℱ$ is the sheaf $\widetilde{ℱ}$ given by $$\begin{array}{rcl}\widetilde{ℱ}\left(U\right)& =& \left\{s\prime =\left({s\prime }_x\right)\in \prod _{x\in U}{ℱ}_x|\begin{array}{l}\text{if}x\in U\text{then there exists}V\in 𝒯\text{, and}\\ s\in ℱ\left(V\right)\text{such that}x\in V\subseteq U\text{, and}\\ {\left({s\prime }_y\right)}_{y\in V}={\left(s_y\right)}_{y\in V}\end{array}\right\}\\ & =& \left\{s\prime :U\to E|s\prime \text{is continuous and}p\circ \tilde{s}={\mathrm{id}}_U\right\}\end{array}$$ where $p:E\to X$ sends ${ℱ}_x$ to $x$.

We have homomorphisms $$\begin{array}{rcl}ℱ\left(U\right)& \hookrightarrow & \widetilde{ℱ}\left(U\right)\\ s& \mapsto & {\left(s_x\right)}_{x\in U}\text{.}\end{array}$$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)